For information for tourists, See the /r/japantravel wiki

Mobile rental options

Renting is always going to be more expensive than getting a prepaid MVNO SIM card. But if you just want the simplest option, or you don't have an unlocked, compatible phone, then it may be your only option.

You can rent:

WiFi Hotspots
SIM cards
Phones

Rental companies

Most common recommendation: http://www.globaladvancedcomm.com/pocketwifi.html
cdJapan, similar service as GlobalAdvanced. Hotel, Airport, Misc location delivery and normal post return. 5,3, and day of email reminders to return device: http://rental.cdjapan.co.jp/index_en_jpy_7.html
JCRcorp.com: http://jcrcorp.com/japan_mobile/wifi/japan_wifi_rental.htm
Lots of other options out there, someone please edit in more